# Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model

## TL;DR

This paper introduces a hierarchical approach combining adaptive sparse grids and quasi-Monte Carlo methods to significantly improve the efficiency of option pricing under the computationally intensive rough Bergomi model.

## Contribution

The paper presents a novel hierarchical method that couples adaptive sparse grids and quasi-Monte Carlo techniques with advanced error correction for efficient option pricing in the rough Bergomi model.

## Key findings

- Substantial computational gains over standard Monte Carlo methods.
- Effective handling of small Hurst parameter values.
- Demonstrated accuracy across various parameter sets.

## Abstract

The rough Bergomi (rBergomi) model, introduced recently in [5], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet remarkably fits with empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a time-consuming task. To overcome this issue, we have designed a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi-Monte Carlo (QMC). Both techniques are coupled with a Brownian bridge construction and a Richardson extrapolation on the weak error. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.